The form of the partial fraction decomposition of a rational function is given below. x...

Kapalci

Kapalci

Answered

2022-06-25

The form of the partial fraction decomposition of a rational function is given below.
x 3 x 2 26 ( x + 1 ) ( x 2 + 9 ) = A x + 1 + B x + C x 2 + 9
What are the values of A , B and C?
So can someone explain how to get the answer. I'm not sure how I got to the answer but I know B = 0 and C = 1
I also found the indefinite integral, which equals
1 3 tan 1 ( x 3 ) 3 log ( 1 + x ) + C

Answer & Explanation

Lamont Adkins

Lamont Adkins

Expert

2022-06-26Added 11 answers

Multiplying either sides by ( x + 1 ) ( x 2 + 9 ) ,
x 3 x 2 26 = A ( x 2 + 9 ) + ( B x + C ) ( x + 1 )
3 x 2 + x 26 = x 2 ( A + B ) + x ( B + C ) + 9 A + C
Compare the constants and the coefficients of x , x 2 of the above identity to form three simultaneous equations with three unknowns A,B,C
George Bray

George Bray

Expert

2022-06-27Added 12 answers

Multiply by x + 1 and let x 1 to obtain
1 3 26 ( 1 ) 2 + 9 = A A = 3.
Multiply both sides by x, let x , and use A = 3 to obtain
3 = A + B B = 0.
Finally, evaluate at x = 0, and use A = 27 / 9:
26 9 = A + C 9 C = 9 ( 26 9 + 27 9 ) = 1.

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